Power Set: Overview, Questions, Preparation

Power Set is a set of all the subsets of a set. A power set includes the set as well as the null or empty set. Power set contains all the possible combinations of subsets, apart from the set itself.

The Number of Elements of a Power Set

The power set of a set A is denoted as P(A). Given a set has n elements, the power set will have 2n elements. For example, a set with one element, A={a}, has 21=2 elements (n=1). P(A) = { {}, {2} }. So a power set of a single element contains the set itself and the null set.

Let us consider a set A = {x, y, z}. Number of elements for P(A) = 23 = 8

The subsets of set A are:

{} (null/empty set)

{x}

{y}

{z}

{x,y}

{y,z}

{x,z}

{x,y,z}

P(A) = {{}, {x}, {y}, {z}, {x,y}, {y,z}, {x,z}, {x,y,z}}

Explanation of Subsets of a Set

Φ (null set or empty set) is a subset of every set. Also, every set is a subset of itself. Rest of the subsets are a combination of the elements in all possible ways. So, a set

A = {a} has two subsets, set A = {a} and Φ.

Recursive Algorithm

F (e,T) = { X ∪ {e} | X ∈ T }

The above is a recursive algorithm, used to generate a power set of any set X in T containing element x.

So, if Set S = {}, P(S) = { {} }

If not, the algorithm is returned as follows.

Given e to be an element in Set S, T = S {e}

S {e} forming a relative complement of element e in set S. The Power Set is as follows:

P(S) = P(T) ∪ F ( e, P(T))

Power Set and Binomial Theorem

Notation for a Power Set is closely related to Binomial Theorem.

Let’s look at a set S = {a,b,c}

Total elements in the set = 3

Total subsets with one element = 3

Total subsets with two elements = 3

Total null subset = 1

Total subset with 3 elements, i.e., the set itself = 1

Here, we find |2s|to be:

Now let us say that |S| = n then,

This is how the power set and binomial theorem are linked with each other.

Importance and Weightage

Relevance in Class X

Sets form the first Chapter in Class X Mathematics, and Power Sets is a part of the main chapter.

Relevance in Class XI

Sets form the first Chapter in Class XI Mathematics and form the start of Class XI-XII Mathematics. Power Sets is a part of the main chapter.

1. Find Power Set of set X = {3, 9, 11} and total number of elements.
Solution. X = {3,9,11}.

The total number of elements of X = 3.

Total number of elements of P (X) = 2n=3 = 8.

P (X) = { {}, {3}, {9}, {11}, {3,9}, {3,11}, {9,11}, {3,9,11} }

2. How many elements has P(A), if A = φ?

Solution. The total number of elements of a null set is zero.

So, n = 0.

Total number of elements of the power set of a null set = 2n=0 = 1.

The null set itself is the only element of a power set of the null set.

3. How many options will you have for the order if you go to an ice-cream parlor with your group having 6 different ice-cream flavors?

Solution. From nobody ordering anything (all full or don’t like the place?) constituting a null set, to each one having different flavour, thus all being ordered, there are 26 options or 64 options for you to order.

Q: Define Power Set.

Q: How can you define elements of a power set?

Q: Why is there an element in the power set of a null set?

Q: Show the elements of Power Set of {1,2,3,4}

Q: How would you define the cardinality of a power set?